Overlaps

AtCoder

IDarc137_f

Time6000ms

Memory256MB

Difficulty—

We have a bar of length $1$. A point on the bar whose distance from the left end of the bar is $x$ is said to have a coordinate $x$. Snuke will do the operation below $N$ times. * Choose two real numbers $x$ and $y$ uniformly at random from $[0, 1]$. Put a sticker covering the range from the coordinate $\min(x,y)$ to the coordinate $\max(x,y)$. Here, all random choices are independent of each other. Stickers can overlap. The bar is said to be good when no point is covered by $K+1$ or more stickers. Find the probability, modulo $998244353$, of having a good bar after putting $N$ stickers. Definition of a probability modulo $998244353$It can be proved that the sought probability is always a rational number. Additionally, under the Constraints of this problem, when that value is represented as an irreducible fraction $\frac{P}{Q}$, it can be proved that $Q \not \equiv 0 \pmod{998244353}$. Thus, there is a unique integer $R$ such that $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$. Report this $R$. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq K \leq \min(N,10^5)$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $K$ [samples]

Samples

Input #1

2 1

Output #1

332748118

We are to find the probability that the two stickers do not overlap, which is $1/3$.

Input #2

5 3

Output #2

66549624

Input #3

10000 5000

Output #3

642557092

API Response (JSON)

{
  "problem": {
    "name": "Overlaps",
    "description": {
      "content": "We have a bar of length $1$. A point on the bar whose distance from the left end of the bar is $x$ is said to have a coordinate $x$. Snuke will do the operation below $N$ times. *   Choose two real n",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 6000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc137_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a bar of length $1$. A point on the bar whose distance from the left end of the bar is $x$ is said to have a coordinate $x$.\nSnuke will do the operation below $N$ times.\n\n*   Choose two real n...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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